An uncoupled theory of FG nanobeams with the small size effects and its exact solutions

摘要

Due to unphysical coupling induced by the material inhomogeneity, FG (functionally graded) nanobeam problems were formulated in a very complex way so that they cannot be analytically solved. In this paper, an uncoupled theory is proposed for FG nanobeams considering their small size effects. First, with the aid of the neutral axis, the axial displacement is expressed in terms of generalized displacements for FG nanobeams. Based on the nonlocal strain gradient theory, the generalized stresses and strains are accordingly defined and uncoupled constitutive relations are derived. Based on the principle of virtual work, an uncoupled theory is eventually established, including governing equations and boundary conditions. Within the present framework, analytical solutions to FG nanobeams are obtained for the first time for general boundary conditions. These solutions not only re-evaluate the previous results but shed light on the small size effects of FG nanobeams.